Euler s, Fermat s and Wilson s Theorems
|
|
- Jeffry Powers
- 5 years ago
- Views:
Transcription
1 Euler s, Fermat s and Wilson s Theorems R. C. Daileda February 17, Euler s Theorem Consider the following example. Example 1. Find the remainder when is divided by 14. We begin by computing successive powers of 3 modulo 14. The computations can easily be carried out in our heads since at each line we can always be sure we re never dealing with a number larger than 13, and the next line is simply obtained by multiplying the result of the previous line by (mod 14), (mod 14), (mod 14), (mod 14), (mod 14). Why did we stop here? Divide the real exponent we care about, 103, by 6: 103 = We then have = = (3 6 ) (mod 14) which means the remainder is 3. We took advantage of the fact that 3 6 1(mod 14) and that multiplication by 1 is trivial to convert the problem of reducing modulo 14 to the much more reasonable task of reducing the exponent 103 modulo 6. The reason we successively raised 3 to powers was to find the smallest exponent (namely 6) that would help us out. It turns out that what happened with the powers of 3 in the preceding example generalizes nicely to arbitrary moduli. As usual, we will introduce an abstract notion first and then return to congruences by applying it to Z/nZ. Definition 1. Let G be a group, g G. If the set {n n N and g n = e} is nonempty, we define the order of g to be its least element. Otherwise we say the order of g is infinite. We denote the order of g by ord(g). So, ord(g) is the least n N so that g n = e, if such an integer exists, and is infinite otherwise. Example 2. 1
2 ord(g) = 1 if and only if g = e. In the example above, we see that ord(3) = 6 in (Z/14Z). For n Z, ord(n) is infinite if n 0 (ord(0) = 1 by our first observation). This is because if n 0 and k N, then kn 1 so that kn 0 (remember Z is additive, not multiplicative, so we don t use exponents). On the other hand, every element of Z/nZ has finite order, since n(a+nz) = na+nz = 0 + nz. Lemma 1. If G is a finite group and g G, then ord(g) is finite. Proof. Since the list g, g 2, g 3, g 4, cannot contain infinitely many distinct elements, there exist exponents i > j so that g i = g j. Cancelling we obtain g i j = e. Since i j N, this proves g has finite order. 1 Corollary 1. Let n N. Every element of (Z/nZ) has finite order. Remark 1. Z/nZ possesses two binary operations (although is only a group under one of them), hence has two notions of order. By caveat, from now on when we speak of order in Z/nZ we will always mean multiplicative order (see below). When n is understood, we will write ord(a) for ord(a + nz) in (Z/nZ). We will call it the order of a modulo n. If g n = e in G, then g nm = (g n ) m = e m = e for all m Z. Proposition 1. Let G be a group and g G an element of finite order. Then {n n Z and g n = e} = ord(g)z. Proof. The final remark above shows that we have the containment. For the other, let n belong to the left hand side. Use the division algorithm to write n = q ord(g) + r with 0 r < ord(g). Then e = g n = (g ord(g) ) q g r = e q g r = g r. This contradicts the minimality of ord(g) unless r = 0, so that n ord(g)z, as claimed. In words, the proposition states that the only exponents that annihilate a group element are multiples of its order. Given a finite group G, a somewhat natural question to ask is if there is a common power that will annihilate every element. According to the proposition, an obvious, although probably cumbersome to compute, choice is the least common multiple of all of the orders of the elements of G. But there is a more natural choice, as the next result indicates. 1 Note that we are not claiming that ord(g) = i j, just that the set of positive powers of g equal to e is nonempty. 2
3 Theorem 1. Let G be a finite abelian group, g G. Then In particular, ord(g) divides G. g G = e. Proof. Recall from HW that the left translation L g : G G, given by x gx, is a bijection. Hence 2 L g (x) = x e = g G, x G x = x G by cancellation of the product from both sides. x Ggx = g G x G Remark 2. The preceding result holds for all finite groups G, although the proof, while still very elegant, is not quite as simple. It requires a fundamental group-theoretic result known as Lagrange s Theorem. We are almost ready to state Euler s Theorem, which is simply the theorem above couched in the language of congruences. To simplify its statement somewhat, and because it will later be of independent interest, we introduce our first number-theoretic function. Definition 2. The Euler ϕ function (or totient function), ϕ : N N, is defined by ϕ(n) = (Z/nZ) = {a N 1 a n and (a, n) = 1}. Remark 3. Since Z/Z consists of a single element, it isn t technically a ring, so we can t talk about its unit group, meaning ϕ(1) = (Z/Z) doesn t make sense. However, the alternate characterization given above does make sense when n = 1 and shows that we should take ϕ(1) = 1. Example 3. n (Z/nZ) ϕ(n) {1} 1 3 {1, 2} 2 4 {1, 3} 2 5 {1, 2, 3, 4} 4 6 {1, 5} 2 7 {1, 2, 3, 4, 5, 6} 6 8 {1, 3, 5, 7} 4 9 {1, 2, 4, 5, 7, 8} 6 10 {1, 3, 7, 9} 4 11 {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} {1, 5, 7, 11} 4 2 The products below are well-defined since G is abelian so we need not specify an order of multiplication. 3
4 Remark 4. Carmichael s conjecture is that ϕ is always at least 2-to-1, i.e. given any n N there exists an m N, m n, so that ϕ(m) = ϕ(n). Corollary 2 (Euler s Theorem). Let n N and a Z. If (a, n) = 1, then a ϕ(n) 1 (mod n). In particular, the order of a modulo n divides ϕ(n). Proof. If (a, n) = 1, then a + nz (Z/nZ). Since ϕ(n) = (Z/nZ), the theorem implies that (a + nz) ϕ(n) = 1 + nz a ϕ(n) + nz = 1 + nz a ϕ(n) 1 (mod n). The equation (a + nz) ϕ(n) = 1 + nz also shows that the order of a modulo n divides ϕ(n), since it implies the order of a + nz in (Z/nZ) divides ϕ(n). Remark 5. Note that if n N, a Z with (a, n) = 1 and s t(mod ϕ(n)), then a s a t (mod n). Indeed, we have s = t + k ϕ(n) for some k so that by Euler s theorem a s = (a ϕ(n) ) k a t 1 k a t a t (mod n). Hence we can reduce the exponent modulo ϕ(n) in order to reduce the overall power modulo n. Example 4. Find the remainder when is divided by 20. Rather than compute the order of 7 modulo 20 as we did with our initial example, we use Euler s theorem as a substitute. Since ϕ(20) = 8 and (mod 8), we have (mod 20) 9 (mod 20) so that the remainder is 9. Remark 6. In the preceding example, the actual order of 7 modulo 20 is 4, which probably is no harder to compute directly than ϕ(20). However, once we have some rules for computing ϕ(n) we will see that it is frequently much easier to find than the order of a given element and therefore much more convenient to work with in problems like the one above. We conclude this section with Fermat s Little Theorem. Historically Fermat s theorem preceded Euler s, and the latter served to generalize the former. However, in our presentation it is more natural to simply present Fermat s theorem as a special case of Euler s result. Nonetheless, it is a valuable result to keep in mind. Corollary 3 (Fermat s Little Theorem). Let p be a prime and a Z. If p a, then a p 1 1 (mod p). Proof. Since p is prime, ϕ(p) = p 1 and p a implies (a, p) = 1. The result then follows immediately from Euler s theorem. 4
5 2 Wilson s Theorem Let G be an abelian group and consider the product that occurred in the proof of Theorem 1: P = x G x. Notice that if p is prime and G = (Z/pZ), then this product is just the congruence class (p 1)! + pz. It is natural to ask how this product depends on the group G. To answer this question we define G(2) = {g G g 2 = e}. Note that G(2) consists of the identity element and the elements of G of order 2 (if there are any). Remark 7. Let G be a group and suppose g G has order 2. Then g 2 = e which implies g 1 = g. So G(2) can also be thought of as the set of all elements of G that are their own inverses. For an abelian group G, the subset G(2) is closed under the binary operation on G and itself satisfies the axioms of a group, i.e. is a subgroup of G. It is called the 2-torsion subgroup of G. Example 5. If G = (Z/nZ), then x+nz (Z/nZ) belongs to G(2) if and only if x 2 +nz = 1+nZ if and only if x 2 1(mod n). In the homework you found that x 2 1 (mod 35) if and only if x ±1, ±6(mod 35). Hence, if G = Z/35Z, then G(2) = {±1 + 35Z, ±6 + 35Z}. If p is prime, then x 2 1(mod p) if and only if p x 2 1 = (x 1)(x + 1) if and only if p (x 1) or p (x + 1) if and only if x 1(mod p) or x 1(mod p). Hence, for the group G = (Z/pZ) G(2) = {±1 + pz}. We now prove our primary abstract result. Theorem 2. Let G be an abelian group. Then P = x G x = x G(2) Proof. In the product P pair each element with its inverse, when its inverse is distinct from itself. This leaves behind only those elements that are their own inverses. x. 5
6 Corollary 4 (Wilson s Theorem). If p is prime, then (p 1)! 1 (mod p). Proof. Let G = (Z/pZ). We have already observed that x = (p 1)! + pz. We have also seen that G(2) = {±1 + pz} so that x = (1 + pz)( 1 + pz) = 1 + pz. x G(2) x G By the theorem, we therefore have (p 1)! + pz = 1 + pz which is equivalent to the statement of the corollary. Remark 8. The converse of Wilson s theorem is also true. So given n N, determining the congruence class of (n 1)! modulo n serves as a primality test for n. Unfortunately it is extremely inefficient. Example 6. Find the remainder when 99! is divided by 103. We note that 103 is prime and so by Wilson s theorem Hence 1 102! 99! !( 3)( 2)( 1) 99! 3 2 (mod 103). 99! (mod 103) where the the negative exponents indicate inverses modulo 103 (i.e. in (Z/103Z) ). The extended Euclidean algorithm takes a single step for both 2 and 3 and immediately yields the inverses 52 and 69, respectively. Hence 99! (mod 103) so that the remainder is 86. Given that 99! has 156 digits, the fact that we were able to compute its remainder so easily (by hand!) demonstrates the utility of Wilson s theorem. 6
MATH 25 CLASS 21 NOTES, NOV Contents. 2. Subgroups 2 3. Isomorphisms 4
MATH 25 CLASS 21 NOTES, NOV 7 2011 Contents 1. Groups: definition 1 2. Subgroups 2 3. Isomorphisms 4 1. Groups: definition Even though we have been learning number theory without using any other parts
More informationAn integer p is prime if p > 1 and p has exactly two positive divisors, 1 and p.
Chapter 6 Prime Numbers Part VI of PJE. Definition and Fundamental Results Definition. (PJE definition 23.1.1) An integer p is prime if p > 1 and p has exactly two positive divisors, 1 and p. If n > 1
More informationKnow the Well-ordering principle: Any set of positive integers which has at least one element contains a smallest element.
The first exam will be on Monday, June 8, 202. The syllabus will be sections. and.2 in Lax, and the number theory handout found on the class web site, plus the handout on the method of successive squaring
More informationDiscrete Mathematics with Applications MATH236
Discrete Mathematics with Applications MATH236 Dr. Hung P. Tong-Viet School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal Pietermaritzburg Campus Semester 1, 2013 Tong-Viet
More informationCHAPTER 6. Prime Numbers. Definition and Fundamental Results
CHAPTER 6 Prime Numbers Part VI of PJE. Definition and Fundamental Results 6.1. Definition. (PJE definition 23.1.1) An integer p is prime if p > 1 and the only positive divisors of p are 1 and p. If n
More informationMATH 361: NUMBER THEORY FOURTH LECTURE
MATH 361: NUMBER THEORY FOURTH LECTURE 1. Introduction Everybody knows that three hours after 10:00, the time is 1:00. That is, everybody is familiar with modular arithmetic, the usual arithmetic of the
More informationWilson s Theorem and Fermat s Little Theorem
Wilson s Theorem and Fermat s Little Theorem Wilson stheorem THEOREM 1 (Wilson s Theorem): (p 1)! 1 (mod p) if and only if p is prime. EXAMPLE: We have (2 1)!+1 = 2 (3 1)!+1 = 3 (4 1)!+1 = 7 (5 1)!+1 =
More informationLecture notes: Algorithms for integers, polynomials (Thorsten Theobald)
Lecture notes: Algorithms for integers, polynomials (Thorsten Theobald) 1 Euclid s Algorithm Euclid s Algorithm for computing the greatest common divisor belongs to the oldest known computing procedures
More informationALGEBRA I (LECTURE NOTES 2017/2018) LECTURE 9 - CYCLIC GROUPS AND EULER S FUNCTION
ALGEBRA I (LECTURE NOTES 2017/2018) LECTURE 9 - CYCLIC GROUPS AND EULER S FUNCTION PAVEL RŮŽIČKA 9.1. Congruence modulo n. Let us have a closer look at a particular example of a congruence relation on
More informationThe Chinese Remainder Theorem
The Chinese Remainder Theorem R. C. Daileda February 19, 2018 1 The Chinese Remainder Theorem We begin with an example. Example 1. Consider the system of simultaneous congruences x 3 (mod 5), x 2 (mod
More informationALGEBRA. 1. Some elementary number theory 1.1. Primes and divisibility. We denote the collection of integers
ALGEBRA CHRISTIAN REMLING 1. Some elementary number theory 1.1. Primes and divisibility. We denote the collection of integers by Z = {..., 2, 1, 0, 1,...}. Given a, b Z, we write a b if b = ac for some
More informationApplied Cryptography and Computer Security CSE 664 Spring 2018
Applied Cryptography and Computer Security Lecture 12: Introduction to Number Theory II Department of Computer Science and Engineering University at Buffalo 1 Lecture Outline This time we ll finish the
More information7.2 Applications of Euler s and Fermat s Theorem.
7.2 Applications of Euler s and Fermat s Theorem. i) Finding and using inverses. From Fermat s Little Theorem we see that if p is prime and p a then a p 1 1 mod p, or equivalently a p 2 a 1 mod p. This
More informationD-MATH Algebra II FS18 Prof. Marc Burger. Solution 26. Cyclotomic extensions.
D-MAH Algebra II FS18 Prof. Marc Burger Solution 26 Cyclotomic extensions. In the following, ϕ : Z 1 Z 0 is the Euler function ϕ(n = card ((Z/nZ. For each integer n 1, we consider the n-th cyclotomic polynomial
More information2 More on Congruences
2 More on Congruences 2.1 Fermat s Theorem and Euler s Theorem definition 2.1 Let m be a positive integer. A set S = {x 0,x 1,,x m 1 x i Z} is called a complete residue system if x i x j (mod m) whenever
More informationChapter 5. Modular arithmetic. 5.1 The modular ring
Chapter 5 Modular arithmetic 5.1 The modular ring Definition 5.1. Suppose n N and x, y Z. Then we say that x, y are equivalent modulo n, and we write x y mod n if n x y. It is evident that equivalence
More informationNUMBER SYSTEMS. Number theory is the study of the integers. We denote the set of integers by Z:
NUMBER SYSTEMS Number theory is the study of the integers. We denote the set of integers by Z: Z = {..., 3, 2, 1, 0, 1, 2, 3,... }. The integers have two operations defined on them, addition and multiplication,
More informationNumber Theory and Group Theoryfor Public-Key Cryptography
Number Theory and Group Theory for Public-Key Cryptography TDA352, DIT250 Wissam Aoudi Chalmers University of Technology November 21, 2017 Wissam Aoudi Number Theory and Group Theoryfor Public-Key Cryptography
More information5 Group theory. 5.1 Binary operations
5 Group theory This section is an introduction to abstract algebra. This is a very useful and important subject for those of you who will continue to study pure mathematics. 5.1 Binary operations 5.1.1
More informationMATH 310: Homework 7
1 MATH 310: Homework 7 Due Thursday, 12/1 in class Reading: Davenport III.1, III.2, III.3, III.4, III.5 1. Show that x is a root of unity modulo m if and only if (x, m 1. (Hint: Use Euler s theorem and
More informationORDERS OF ELEMENTS IN A GROUP
ORDERS OF ELEMENTS IN A GROUP KEITH CONRAD 1. Introduction Let G be a group and g G. We say g has finite order if g n = e for some positive integer n. For example, 1 and i have finite order in C, since
More informationThe Chinese Remainder Theorem
Chapter 5 The Chinese Remainder Theorem 5.1 Coprime moduli Theorem 5.1. Suppose m, n N, and gcd(m, n) = 1. Given any remainders r mod m and s mod n we can find N such that N r mod m and N s mod n. Moreover,
More informationAN ALGEBRA PRIMER WITH A VIEW TOWARD CURVES OVER FINITE FIELDS
AN ALGEBRA PRIMER WITH A VIEW TOWARD CURVES OVER FINITE FIELDS The integers are the set 1. Groups, Rings, and Fields: Basic Examples Z := {..., 3, 2, 1, 0, 1, 2, 3,...}, and we can add, subtract, and multiply
More information2x 1 7. A linear congruence in modular arithmetic is an equation of the form. Why is the solution a set of integers rather than a unique integer?
Chapter 3: Theory of Modular Arithmetic 25 SECTION C Solving Linear Congruences By the end of this section you will be able to solve congruence equations determine the number of solutions find the multiplicative
More information1 Structure of Finite Fields
T-79.5501 Cryptology Additional material September 27, 2005 1 Structure of Finite Fields This section contains complementary material to Section 5.2.3 of the text-book. It is not entirely self-contained
More informationChapter 9 Mathematics of Cryptography Part III: Primes and Related Congruence Equations
Chapter 9 Mathematics of Cryptography Part III: Primes and Related Congruence Equations Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 9.1 Chapter 9 Objectives
More informationA Generalization of Wilson s Theorem
A Generalization of Wilson s Theorem R. Andrew Ohana June 3, 2009 Contents 1 Introduction 2 2 Background Algebra 2 2.1 Groups................................. 2 2.2 Rings.................................
More informationECEN 5022 Cryptography
Elementary Algebra and Number Theory University of Colorado Spring 2008 Divisibility, Primes Definition. N denotes the set {1, 2, 3,...} of natural numbers and Z denotes the set of integers {..., 2, 1,
More information1. multiplication is commutative and associative;
Chapter 4 The Arithmetic of Z In this chapter, we start by introducing the concept of congruences; these are used in our proof (going back to Gauss 1 ) that every integer has a unique prime factorization.
More informationMathematics for Cryptography
Mathematics for Cryptography Douglas R. Stinson David R. Cheriton School of Computer Science University of Waterloo Waterloo, Ontario, N2L 3G1, Canada March 15, 2016 1 Groups and Modular Arithmetic 1.1
More informationMath 324, Fall 2011 Assignment 7 Solutions. 1 (ab) γ = a γ b γ mod n.
Math 324, Fall 2011 Assignment 7 Solutions Exercise 1. (a) Suppose a and b are both relatively prime to the positive integer n. If gcd(ord n a, ord n b) = 1, show ord n (ab) = ord n a ord n b. (b) Let
More informationPart II. Number Theory. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 Paper 3, Section I 1G 70 Explain what is meant by an Euler pseudoprime and a strong pseudoprime. Show that 65 is an Euler
More informationThe number of ways to choose r elements (without replacement) from an n-element set is. = r r!(n r)!.
The first exam will be on Friday, September 23, 2011. The syllabus will be sections 0.1 through 0.4 and 0.6 in Nagpaul and Jain, and the corresponding parts of the number theory handout found on the class
More information2 Lecture 2: Logical statements and proof by contradiction Lecture 10: More on Permutations, Group Homomorphisms 31
Contents 1 Lecture 1: Introduction 2 2 Lecture 2: Logical statements and proof by contradiction 7 3 Lecture 3: Induction and Well-Ordering Principle 11 4 Lecture 4: Definition of a Group and examples 15
More informationContest Number Theory
Contest Number Theory Andre Kessler December 7, 2008 Introduction Number theory is one of the core subject areas of mathematics. It can be somewhat loosely defined as the study of the integers. Unfortunately,
More informationCorollary 4.2 (Pepin s Test, 1877). Let F k = 2 2k + 1, the kth Fermat number, where k 1. Then F k is prime iff 3 F k 1
4. Primality testing 4.1. Introduction. Factorisation is concerned with the problem of developing efficient algorithms to express a given positive integer n > 1 as a product of powers of distinct primes.
More informationSOLUTIONS TO PROBLEM SET 1. Section = 2 3, 1. n n + 1. k(k + 1) k=1 k(k + 1) + 1 (n + 1)(n + 2) n + 2,
SOLUTIONS TO PROBLEM SET 1 Section 1.3 Exercise 4. We see that 1 1 2 = 1 2, 1 1 2 + 1 2 3 = 2 3, 1 1 2 + 1 2 3 + 1 3 4 = 3 4, and is reasonable to conjecture n k=1 We will prove this formula by induction.
More informationK. Ireland, M. Rosen A Classical Introduction to Modern Number Theory, Springer.
Chapter 1 Number Theory and Algebra 1.1 Introduction Most of the concepts of discrete mathematics belong to the areas of combinatorics, number theory and algebra. In Chapter?? we studied the first area.
More informationDivisibility = 16, = 9, = 2, = 5. (Negative!)
Divisibility 1-17-2018 You probably know that division can be defined in terms of multiplication. If m and n are integers, m divides n if n = mk for some integer k. In this section, I ll look at properties
More informationFoundations of Cryptography
Foundations of Cryptography Ville Junnila viljun@utu.fi Department of Mathematics and Statistics University of Turku 2015 Ville Junnila viljun@utu.fi Lecture 7 1 of 18 Cosets Definition 2.12 Let G be a
More informationIRREDUCIBILITY TESTS IN F p [T ]
IRREDUCIBILITY TESTS IN F p [T ] KEITH CONRAD 1. Introduction Let F p = Z/(p) be a field of prime order. We will discuss a few methods of checking if a polynomial f(t ) F p [T ] is irreducible that are
More informationThe group (Z/nZ) February 17, In these notes we figure out the structure of the unit group (Z/nZ) where n > 1 is an integer.
The group (Z/nZ) February 17, 2016 1 Introduction In these notes we figure out the structure of the unit group (Z/nZ) where n > 1 is an integer. If we factor n = p e 1 1 pe, where the p i s are distinct
More informationNumber Theory A focused introduction
Number Theory A focused introduction This is an explanation of RSA public key cryptography. We will start from first principles, but only the results that are needed to understand RSA are given. We begin
More informationA Guide to Arithmetic
A Guide to Arithmetic Robin Chapman August 5, 1994 These notes give a very brief resumé of my number theory course. Proofs and examples are omitted. Any suggestions for improvements will be gratefully
More informationQ 2.0.2: If it s 5:30pm now, what time will it be in 4753 hours? Q 2.0.3: Today is Wednesday. What day of the week will it be in one year from today?
2 Mod math Modular arithmetic is the math you do when you talk about time on a clock. For example, if it s 9 o clock right now, then it ll be 1 o clock in 4 hours. Clearly, 9 + 4 1 in general. But on a
More informationDiscrete Logarithms. Let s begin by recalling the definitions and a theorem. Let m be a given modulus. Then the finite set
Discrete Logarithms Let s begin by recalling the definitions and a theorem. Let m be a given modulus. Then the finite set Z/mZ = {[0], [1],..., [m 1]} = {0, 1,..., m 1} of residue classes modulo m is called
More informationI216e Discrete Math (for Review)
I216e Discrete Math (for Review) Nov 22nd, 2017 To check your understanding. Proofs of do not appear in the exam. 1 Monoid Let (G, ) be a monoid. Proposition 1 Uniquness of Identity An idenity e is unique,
More informationGroups, Rings, and Finite Fields. Andreas Klappenecker. September 12, 2002
Background on Groups, Rings, and Finite Fields Andreas Klappenecker September 12, 2002 A thorough understanding of the Agrawal, Kayal, and Saxena primality test requires some tools from algebra and elementary
More informationNotes on Primitive Roots Dan Klain
Notes on Primitive Roots Dan Klain last updated March 22, 2013 Comments and corrections are welcome These supplementary notes summarize the presentation on primitive roots given in class, which differed
More informationCongruences and Residue Class Rings
Congruences and Residue Class Rings (Chapter 2 of J. A. Buchmann, Introduction to Cryptography, 2nd Ed., 2004) Shoichi Hirose Faculty of Engineering, University of Fukui S. Hirose (U. Fukui) Congruences
More informationDefinitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations
Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of
More informationLecture 4: Number theory
Lecture 4: Number theory Rajat Mittal IIT Kanpur In the next few classes we will talk about the basics of number theory. Number theory studies the properties of natural numbers and is considered one of
More informationDiscrete Mathematics and Probability Theory Fall 2014 Anant Sahai Homework 5. This homework is due October 6, 2014, at 12:00 noon.
EECS 70 Discrete Mathematics and Probability Theory Fall 2014 Anant Sahai Homework 5 This homework is due October 6, 2014, at 12:00 noon. 1. Modular Arithmetic Lab (continue) Oystein Ore described a puzzle
More informationNumbers, Groups and Cryptography. Gordan Savin
Numbers, Groups and Cryptography Gordan Savin Contents Chapter 1. Euclidean Algorithm 5 1. Euclidean Algorithm 5 2. Fundamental Theorem of Arithmetic 9 3. Uniqueness of Factorization 14 4. Efficiency
More informationNOTES ON FINITE FIELDS
NOTES ON FINITE FIELDS AARON LANDESMAN CONTENTS 1. Introduction to finite fields 2 2. Definition and constructions of fields 3 2.1. The definition of a field 3 2.2. Constructing field extensions by adjoining
More informationA Curious Connection Between Fermat Numbers and Finite Groups
A Curious Connection Between Fermat Numbers and Finite Groups Carrie E. Finch and Lenny Jones 1. INTRODUCTION. In the seventeenth century, Fermat defined the sequence of numbers F n = 2 2n + 1 for n 0,
More informationPREPARATION NOTES FOR NUMBER THEORY PRACTICE WED. OCT. 3,2012
PREPARATION NOTES FOR NUMBER THEORY PRACTICE WED. OCT. 3,2012 0.1. Basic Num. Th. Techniques/Theorems/Terms. Modular arithmetic, Chinese Remainder Theorem, Little Fermat, Euler, Wilson, totient, Euclidean
More informationYALE UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE
YALE UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE CPSC 467a: Cryptography and Computer Security Notes 13 (rev. 2) Professor M. J. Fischer October 22, 2008 53 Chinese Remainder Theorem Lecture Notes 13 We
More informationDefinition 6.1 (p.277) A positive integer n is prime when n > 1 and the only positive divisors are 1 and n. Alternatively
6 Prime Numbers Part VI of PJE 6.1 Fundamental Results Definition 6.1 (p.277) A positive integer n is prime when n > 1 and the only positive divisors are 1 and n. Alternatively D (p) = { p 1 1 p}. Otherwise
More information2x 1 7. A linear congruence in modular arithmetic is an equation of the form. Why is the solution a set of integers rather than a unique integer?
Chapter 3: Theory of Modular Arithmetic 25 SECTION C Solving Linear Congruences By the end of this section you will be able to solve congruence equations determine the number of solutions find the multiplicative
More informationSOLUTIONS Math 345 Homework 6 10/11/2017. Exercise 23. (a) Solve the following congruences: (i) x (mod 12) Answer. We have
Exercise 23. (a) Solve the following congruences: (i) x 101 7 (mod 12) Answer. We have φ(12) = #{1, 5, 7, 11}. Since gcd(7, 12) = 1, we must have gcd(x, 12) = 1. So 1 12 x φ(12) = x 4. Therefore 7 12 x
More information* 8 Groups, with Appendix containing Rings and Fields.
* 8 Groups, with Appendix containing Rings and Fields Binary Operations Definition We say that is a binary operation on a set S if, and only if, a, b, a b S Implicit in this definition is the idea that
More informationNumber Theory Homework.
Number Theory Homewor. 1. The Theorems of Fermat, Euler, and Wilson. 1.1. Fermat s Theorem. The following is a special case of a result we have seen earlier, but as it will come up several times in this
More informationCHAPTER 3. Congruences. Congruence: definitions and properties
CHAPTER 3 Congruences Part V of PJE Congruence: definitions and properties Definition. (PJE definition 19.1.1) Let m > 0 be an integer. Integers a and b are congruent modulo m if m divides a b. We write
More informationDefinition 1.2. Let G be an arbitrary group and g an element of G.
1. Group Theory I Section 2.2 We list some consequences of Lagrange s Theorem for exponents and orders of elements which will be used later, especially in 2.6. Definition 1.1. Let G be an arbitrary group
More informationInstructor: Bobby Kleinberg Lecture Notes, 25 April The Miller-Rabin Randomized Primality Test
Introduction to Algorithms (CS 482) Cornell University Instructor: Bobby Kleinberg Lecture Notes, 25 April 2008 The Miller-Rabin Randomized Primality Test 1 Introduction Primality testing is an important
More informationCommutative Rings and Fields
Commutative Rings and Fields 1-22-2017 Different algebraic systems are used in linear algebra. The most important are commutative rings with identity and fields. Definition. A ring is a set R with two
More informationBasic elements of number theory
Cryptography Basic elements of number theory Marius Zimand 1 Divisibility, prime numbers By default all the variables, such as a, b, k, etc., denote integer numbers. Divisibility a 0 divides b if b = a
More informationBasic elements of number theory
Cryptography Basic elements of number theory Marius Zimand By default all the variables, such as a, b, k, etc., denote integer numbers. Divisibility a 0 divides b if b = a k for some integer k. Notation
More informationAll variables a, b, n, etc are integers unless otherwise stated. Each part of a problem is worth 5 points.
Math 152, Problem Set 2 solutions (2018-01-24) All variables a, b, n, etc are integers unless otherwise stated. Each part of a problem is worth 5 points. 1. Let us look at the following equation: x 5 1
More informationChapter 3. Rings. The basic commutative rings in mathematics are the integers Z, the. Examples
Chapter 3 Rings Rings are additive abelian groups with a second operation called multiplication. The connection between the two operations is provided by the distributive law. Assuming the results of Chapter
More informationMath 210B: Algebra, Homework 4
Math 210B: Algebra, Homework 4 Ian Coley February 5, 2014 Problem 1. Let S be a multiplicative subset in a commutative ring R. Show that the localisation functor R-Mod S 1 R-Mod, M S 1 M, is exact. First,
More informationError Correcting Codes Prof. Dr. P Vijay Kumar Department of Electrical Communication Engineering Indian Institute of Science, Bangalore
(Refer Slide Time: 00:54) Error Correcting Codes Prof. Dr. P Vijay Kumar Department of Electrical Communication Engineering Indian Institute of Science, Bangalore Lecture No. # 05 Cosets, Rings & Fields
More informationLecture 6: Finite Fields
CCS Discrete Math I Professor: Padraic Bartlett Lecture 6: Finite Fields Week 6 UCSB 2014 It ain t what they call you, it s what you answer to. W. C. Fields 1 Fields In the next two weeks, we re going
More informationHomework 10 M 373K by Mark Lindberg (mal4549)
Homework 10 M 373K by Mark Lindberg (mal4549) 1. Artin, Chapter 11, Exercise 1.1. Prove that 7 + 3 2 and 3 + 5 are algebraic numbers. To do this, we must provide a polynomial with integer coefficients
More informationLECTURE NOTES IN CRYPTOGRAPHY
1 LECTURE NOTES IN CRYPTOGRAPHY Thomas Johansson 2005/2006 c Thomas Johansson 2006 2 Chapter 1 Abstract algebra and Number theory Before we start the treatment of cryptography we need to review some basic
More information0 Sets and Induction. Sets
0 Sets and Induction Sets A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. We write a A to denote that a is an element of the set
More informationModular Arithmetic Instructor: Marizza Bailey Name:
Modular Arithmetic Instructor: Marizza Bailey Name: 1. Introduction to Modular Arithmetic If someone asks you what day it is 145 days from now, what would you answer? Would you count 145 days, or find
More informationFERMAT S TEST KEITH CONRAD
FERMAT S TEST KEITH CONRAD 1. Introduction Fermat s little theorem says for prime p that a p 1 1 mod p for all a 0 mod p. A naive extension of this to a composite modulus n 2 would be: for all a 0 mod
More informationQuadratic Congruences, the Quadratic Formula, and Euler s Criterion
Quadratic Congruences, the Quadratic Formula, and Euler s Criterion R. C. Trinity University Number Theory Introduction Let R be a (commutative) ring in which 2 = 1 R + 1 R R. Consider a quadratic equation
More informationDirichlet Characters. Chapter 4
Chapter 4 Dirichlet Characters In this chapter we develop a systematic theory for computing with Dirichlet characters, which are extremely important to computations with modular forms for (at least) two
More informationA polytime proof of correctness of the Rabin-Miller algorithm from Fermat s Little Theorem
A polytime proof of correctness of the Rabin-Miller algorithm from Fermat s Little Theorem Grzegorz Herman and Michael Soltys November 24, 2008 Abstract Although a deterministic polytime algorithm for
More informationMa/CS 6a Class 4: Primality Testing
Ma/CS 6a Class 4: Primality Testing By Adam Sheffer Reminder: Euler s Totient Function Euler s totient φ(n) is defined as follows: Given n N, then φ n = x 1 x < n and GCD x, n = 1. In more words: φ n is
More informationQuasi-reducible Polynomials
Quasi-reducible Polynomials Jacques Willekens 06-Dec-2008 Abstract In this article, we investigate polynomials that are irreducible over Q, but are reducible modulo any prime number. 1 Introduction Let
More informationTheory of Numbers Problems
Theory of Numbers Problems Antonios-Alexandros Robotis Robotis October 2018 1 First Set 1. Find values of x and y so that 71x 50y = 1. 2. Prove that if n is odd, then n 2 1 is divisible by 8. 3. Define
More informationTo hand in: (a) Prove that a group G is abelian (= commutative) if and only if (xy) 2 = x 2 y 2 for all x, y G.
Homework #6. Due Thursday, October 14th Reading: For this homework assignment: Sections 3.3 and 3.4 (up to page 167) Before the class next Thursday: Sections 3.5 and 3.4 (pp. 168-171). Also review the
More informationModular Arithmetic and Elementary Algebra
18.310 lecture notes September 2, 2013 Modular Arithmetic and Elementary Algebra Lecturer: Michel Goemans These notes cover basic notions in algebra which will be needed for discussing several topics of
More informationMath 4400/6400 Homework #8 solutions. 1. Let P be an odd integer (not necessarily prime). Show that modulo 2,
MATH 4400 roblems. Math 4400/6400 Homework # solutions 1. Let P be an odd integer not necessarily rime. Show that modulo, { P 1 0 if P 1, 7 mod, 1 if P 3, mod. Proof. Suose that P 1 mod. Then we can write
More informationWednesday, February 21. Today we will begin Course Notes Chapter 5 (Number Theory).
Wednesday, February 21 Today we will begin Course Notes Chapter 5 (Number Theory). 1 Return to Chapter 5 In discussing Methods of Proof (Chapter 3, Section 2) we introduced the divisibility relation from
More information1 Overview and revision
MTH6128 Number Theory Notes 1 Spring 2018 1 Overview and revision In this section we will meet some of the concerns of Number Theory, and have a brief revision of some of the relevant material from Introduction
More informationMATH 3240Q Introduction to Number Theory Homework 4
If the Sun refused to shine I don t mind I don t mind If the mountains fell in the sea Let it be it ain t me Now if six turned out to be nine Oh I don t mind I don t mind Jimi Hendrix If Six Was Nine from
More informationElementary Algebra Chinese Remainder Theorem Euclidean Algorithm
Elementary Algebra Chinese Remainder Theorem Euclidean Algorithm April 11, 2010 1 Algebra We start by discussing algebraic structures and their properties. This is presented in more depth than what we
More information= 1 2x. x 2 a ) 0 (mod p n ), (x 2 + 2a + a2. x a ) 2
8. p-adic numbers 8.1. Motivation: Solving x 2 a (mod p n ). Take an odd prime p, and ( an) integer a coprime to p. Then, as we know, x 2 a (mod p) has a solution x Z iff = 1. In this case we can suppose
More informationAlgebra Review. Instructor: Laszlo Babai Notes by Vincent Lucarelli and the instructor. June 15, 2001
Algebra Review Instructor: Laszlo Babai Notes by Vincent Lucarelli and the instructor June 15, 2001 1 Groups Definition 1.1 A semigroup (G, ) is a set G with a binary operation such that: Axiom 1 ( a,
More informationLecture 2. The Euclidean Algorithm and Numbers in Other Bases
Lecture 2. The Euclidean Algorithm and Numbers in Other Bases At the end of Lecture 1, we gave formulas for the greatest common divisor GCD (a, b), and the least common multiple LCM (a, b) of two integers
More informationCPSC 467: Cryptography and Computer Security
CPSC 467: Cryptography and Computer Security Michael J. Fischer Lecture 9 September 30, 2015 CPSC 467, Lecture 9 1/47 Fast Exponentiation Algorithms Number Theory Needed for RSA Elementary Number Theory
More information6 Cosets & Factor Groups
6 Cosets & Factor Groups The course becomes markedly more abstract at this point. Our primary goal is to break apart a group into subsets such that the set of subsets inherits a natural group structure.
More informationCPSC 467b: Cryptography and Computer Security
CPSC 467b: Cryptography and Computer Security Michael J. Fischer Lecture 8 February 1, 2012 CPSC 467b, Lecture 8 1/42 Number Theory Needed for RSA Z n : The integers mod n Modular arithmetic GCD Relatively
More informationI Foundations Of Divisibility And Congruence 1
Contents I Foundations Of Divisibility And Congruence 1 1 Divisibility 3 1.1 Definitions............................. 3 1.2 Properties Of Divisibility..................... 5 1.3 Some Basic Combinatorial
More information